state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Module.Oriented ℝ V (Fin 2)
s : AffineSubspace ℝ P
p₁ p₂ p₃ p₄ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃p₄ : SOppSide s p₃ p₄
hp₁p₃ : p₁ ≠ p₃
⊢ Real.Ang... | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geo... | rw [← (hp₃p₄.symm.trans (sOppSide_pointReflection hp₁ hp₃p₄.left_not_mem)).oangle_sign_eq hp₁ hp₂,
← oangle_rotate_sign p₁, ← oangle_rotate_sign p₁, oangle_swap₁₃_sign,
(sbtw_pointReflection_of_ne ℝ hp₁p₃).symm.oangle_sign_eq _] | /-- Given two points in an affine subspace, the angles between those two points at two other
points on opposite sides of that subspace have opposite signs. -/
theorem _root_.AffineSubspace.SOppSide.oangle_sign_eq_neg {s : AffineSubspace ℝ P} {p₁ p₂ p₃ p₄ : P}
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃p₄ : s.SOppSide p₃ p₄)... | Mathlib.Geometry.Euclidean.Angle.Oriented.Affine.829_0.cv9h80rIUv6Ug42 | /-- Given two points in an affine subspace, the angles between those two points at two other
points on opposite sides of that subspace have opposite signs. -/
theorem _root_.AffineSubspace.SOppSide.oangle_sign_eq_neg {s : AffineSubspace ℝ P} {p₁ p₂ p₃ p₄ : P}
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃p₄ : s.SOppSide p₃ p₄)... | Mathlib_Geometry_Euclidean_Angle_Oriented_Affine |
X : Type u_1
s : Set X
⊢ (some '' s)ᶜ = some '' sᶜ ∪ {∞} | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [coe_injective.compl_image_eq, compl_range_coe] | theorem compl_image_coe (s : Set X) : ((↑) '' s : Set (OnePoint X))ᶜ = (↑) '' sᶜ ∪ {∞} := by
| Mathlib.Topology.Compactification.OnePoint.140_0.tbRhoScp8wX1EEx | theorem compl_image_coe (s : Set X) : ((↑) '' s : Set (OnePoint X))ᶜ = (↑) '' sᶜ ∪ {∞} | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
x : OnePoint X
⊢ x ≠ ∞ ↔ ∃ y, ↑y = x | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | induction x using OnePoint.rec | theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x := by
| Mathlib.Topology.Compactification.OnePoint.144_0.tbRhoScp8wX1EEx | theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x | Mathlib_Topology_Compactification_OnePoint |
case h₁
X : Type u_1
⊢ ∞ ≠ ∞ ↔ ∃ y, ↑y = ∞ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp | theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x := by
induction x using OnePoint.rec <;> | Mathlib.Topology.Compactification.OnePoint.144_0.tbRhoScp8wX1EEx | theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x | Mathlib_Topology_Compactification_OnePoint |
case h₂
X : Type u_1
x✝ : X
⊢ ↑x✝ ≠ ∞ ↔ ∃ y, ↑y = ↑x✝ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp | theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x := by
induction x using OnePoint.rec <;> | Mathlib.Topology.Compactification.OnePoint.144_0.tbRhoScp8wX1EEx | theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
x : OnePoint X
⊢ x ∉ range some ↔ x = ∞ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff] | theorem not_mem_range_coe_iff {x : OnePoint X} : x ∉ range some ↔ x = ∞ := by
| Mathlib.Topology.Compactification.OnePoint.152_0.tbRhoScp8wX1EEx | theorem not_mem_range_coe_iff {x : OnePoint X} : x ∉ range some ↔ x = ∞ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
⊢ some ⁻¹' {∞} = ∅ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | ext | @[simp]
theorem coe_preimage_infty : ((↑) : X → OnePoint X) ⁻¹' {∞} = ∅ := by
| Mathlib.Topology.Compactification.OnePoint.164_0.tbRhoScp8wX1EEx | @[simp]
theorem coe_preimage_infty : ((↑) : X → OnePoint X) ⁻¹' {∞} = ∅ | Mathlib_Topology_Compactification_OnePoint |
case h
X : Type u_1
x✝ : X
⊢ x✝ ∈ some ⁻¹' {∞} ↔ x✝ ∈ ∅ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp | @[simp]
theorem coe_preimage_infty : ((↑) : X → OnePoint X) ⁻¹' {∞} = ∅ := by
ext
| Mathlib.Topology.Compactification.OnePoint.164_0.tbRhoScp8wX1EEx | @[simp]
theorem coe_preimage_infty : ((↑) : X → OnePoint X) ⁻¹' {∞} = ∅ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
⊢ (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) univ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by | Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set (OnePoint X)
⊢ (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) s →
(fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) t →
(fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) (s ∩ t) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rintro ⟨hms, hs⟩ ⟨hmt, ht⟩ | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
| Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set (OnePoint X)
hms : ∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ
hs : IsOpen (some ⁻¹' s)
hmt : ∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ
ht : IsOpen (some ⁻¹' t)
⊢ (∞ ∈ s ∩ t → IsCompact (some ⁻¹' (s ∩ t))ᶜ) ∧ IsOpen (some ⁻¹' (s ∩ t)) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | refine' ⟨_, hs.inter ht⟩ | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
| Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set (OnePoint X)
hms : ∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ
hs : IsOpen (some ⁻¹' s)
hmt : ∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ
ht : IsOpen (some ⁻¹' t)
⊢ ∞ ∈ s ∩ t → IsCompact (some ⁻¹' (s ∩ t))ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rintro ⟨hms', hmt'⟩ | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
| Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
case intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set (OnePoint X)
hms : ∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ
hs : IsOpen (some ⁻¹' s)
hmt : ∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ
ht : IsOpen (some ⁻¹' t)
hms' : ∞ ∈ s
hmt' : ∞ ∈ t
⊢ IsCompact (some ⁻¹' (s ∩ t))ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simpa [compl_inter] using (hms hms').union (hmt hmt') | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
rintro ⟨hms', hmt'⟩
| Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) t
⊢ (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) (⋃₀ S) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | suffices IsOpen ((↑) ⁻¹' ⋃₀ S : Set X) by
refine' ⟨_, this⟩
rintro ⟨s, hsS : s ∈ S, hs : ∞ ∈ s⟩
refine' IsCompact.of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl _
exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS) | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
rintro ⟨hms', hmt'⟩
simpa [compl_inter] using... | Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) t
this : IsOpen (some ⁻¹' ⋃₀ S)
⊢ (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) (⋃₀ S) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | refine' ⟨_, this⟩ | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
rintro ⟨hms', hmt'⟩
simpa [compl_inter] using... | Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) t
this : IsOpen (some ⁻¹' ⋃₀ S)
⊢ ∞ ∈ ⋃₀ S → IsCompact (some ⁻¹' ⋃₀ S)ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rintro ⟨s, hsS : s ∈ S, hs : ∞ ∈ s⟩ | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
rintro ⟨hms', hmt'⟩
simpa [compl_inter] using... | Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) t
this : IsOpen (some ⁻¹' ⋃₀ S)
s : Set (OnePoint X)
hsS : s ∈ S
hs : ∞ ∈ s
⊢ IsCompact (some ⁻¹' ⋃₀ S)ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | refine' IsCompact.of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl _ | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
rintro ⟨hms', hmt'⟩
simpa [compl_inter] using... | Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) t
this : IsOpen (some ⁻¹' ⋃₀ S)
s : Set (OnePoint X)
hsS : s ∈ S
hs : ∞ ∈ s
⊢ (some ⁻¹' ⋃₀ S)ᶜ ⊆ (some ⁻¹' s)ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS) | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
rintro ⟨hms', hmt'⟩
simpa [compl_inter] using... | Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) t
⊢ IsOpen (some ⁻¹' ⋃₀ S) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [preimage_sUnion] | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
rintro ⟨hms', hmt'⟩
simpa [compl_inter] using... | Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set (OnePoint X))
ho : ∀ t ∈ S, (fun s => (∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ) ∧ IsOpen (some ⁻¹' s)) t
⊢ IsOpen (⋃ t ∈ S, some ⁻¹' t) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact isOpen_biUnion fun s hs => (ho s hs).2 | instance : TopologicalSpace (OnePoint X) where
IsOpen s := (∞ ∈ s → IsCompact (((↑) : X → OnePoint X) ⁻¹' s)ᶜ) ∧
IsOpen (((↑) : X → OnePoint X) ⁻¹' s)
isOpen_univ := by simp
isOpen_inter s t := by
rintro ⟨hms, hs⟩ ⟨hmt, ht⟩
refine' ⟨_, hs.inter ht⟩
rintro ⟨hms', hmt'⟩
simpa [compl_inter] using... | Mathlib.Topology.Compactification.OnePoint.186_0.tbRhoScp8wX1EEx | instance : TopologicalSpace (OnePoint X) where
IsOpen s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
h : ∞ ∈ s
⊢ IsOpen s ↔ IsCompact (some ⁻¹' s)ᶜ ∧ IsOpen (some ⁻¹' s) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [isOpen_def, h] | theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
IsOpen s ↔ IsCompact ((↑) ⁻¹' s : Set X)ᶜ ∧ IsOpen ((↑) ⁻¹' s : Set X) := by
| Mathlib.Topology.Compactification.OnePoint.211_0.tbRhoScp8wX1EEx | theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
IsOpen s ↔ IsCompact ((↑) ⁻¹' s : Set X)ᶜ ∧ IsOpen ((↑) ⁻¹' s : Set X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
h : ∞ ∈ s
⊢ IsOpen s ↔ IsClosed (some ⁻¹' s)ᶜ ∧ IsCompact (some ⁻¹' s)ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp only [isOpen_iff_of_mem' h, isClosed_compl_iff, and_comm] | theorem isOpen_iff_of_mem (h : ∞ ∈ s) :
IsOpen s ↔ IsClosed ((↑) ⁻¹' s : Set X)ᶜ ∧ IsCompact ((↑) ⁻¹' s : Set X)ᶜ := by
| Mathlib.Topology.Compactification.OnePoint.216_0.tbRhoScp8wX1EEx | theorem isOpen_iff_of_mem (h : ∞ ∈ s) :
IsOpen s ↔ IsClosed ((↑) ⁻¹' s : Set X)ᶜ ∧ IsCompact ((↑) ⁻¹' s : Set X)ᶜ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
h : ∞ ∉ s
⊢ IsOpen s ↔ IsOpen (some ⁻¹' s) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [isOpen_def, h] | theorem isOpen_iff_of_not_mem (h : ∞ ∉ s) : IsOpen s ↔ IsOpen ((↑) ⁻¹' s : Set X) := by
| Mathlib.Topology.Compactification.OnePoint.221_0.tbRhoScp8wX1EEx | theorem isOpen_iff_of_not_mem (h : ∞ ∉ s) : IsOpen s ↔ IsOpen ((↑) ⁻¹' s : Set X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
h : ∞ ∈ s
⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | have : ∞ ∉ sᶜ := fun H => H h | theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) := by
| Mathlib.Topology.Compactification.OnePoint.225_0.tbRhoScp8wX1EEx | theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
h : ∞ ∈ s
this : ∞ ∉ sᶜ
⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [← isOpen_compl_iff, isOpen_iff_of_not_mem this, ← isOpen_compl_iff, preimage_compl] | theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) := by
have : ∞ ∉ sᶜ := fun H => H h
| Mathlib.Topology.Compactification.OnePoint.225_0.tbRhoScp8wX1EEx | theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
h : ∞ ∉ s
⊢ IsClosed s ↔ IsClosed (some ⁻¹' s) ∧ IsCompact (some ⁻¹' s) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [← isOpen_compl_iff, isOpen_iff_of_mem (mem_compl h), ← preimage_compl, compl_compl] | theorem isClosed_iff_of_not_mem (h : ∞ ∉ s) :
IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) ∧ IsCompact ((↑) ⁻¹' s : Set X) := by
| Mathlib.Topology.Compactification.OnePoint.230_0.tbRhoScp8wX1EEx | theorem isClosed_iff_of_not_mem (h : ∞ ∉ s) :
IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) ∧ IsCompact ((↑) ⁻¹' s : Set X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s✝ : Set (OnePoint X)
t s : Set X
⊢ IsOpen (some '' s) ↔ IsOpen s | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [isOpen_iff_of_not_mem infty_not_mem_image_coe, preimage_image_eq _ coe_injective] | @[simp]
theorem isOpen_image_coe {s : Set X} : IsOpen ((↑) '' s : Set (OnePoint X)) ↔ IsOpen s := by
| Mathlib.Topology.Compactification.OnePoint.235_0.tbRhoScp8wX1EEx | @[simp]
theorem isOpen_image_coe {s : Set X} : IsOpen ((↑) '' s : Set (OnePoint X)) ↔ IsOpen s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s✝ : Set (OnePoint X)
t s : Set X
⊢ IsOpen (some '' s)ᶜ ↔ IsClosed s ∧ IsCompact s | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [isOpen_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective] | theorem isOpen_compl_image_coe {s : Set X} :
IsOpen ((↑) '' s : Set (OnePoint X))ᶜ ↔ IsClosed s ∧ IsCompact s := by
| Mathlib.Topology.Compactification.OnePoint.240_0.tbRhoScp8wX1EEx | theorem isOpen_compl_image_coe {s : Set X} :
IsOpen ((↑) '' s : Set (OnePoint X))ᶜ ↔ IsClosed s ∧ IsCompact s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s✝ : Set (OnePoint X)
t s : Set X
⊢ ∞ ∈ (some '' s)ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact infty_not_mem_image_coe | theorem isOpen_compl_image_coe {s : Set X} :
IsOpen ((↑) '' s : Set (OnePoint X))ᶜ ↔ IsClosed s ∧ IsCompact s := by
rw [isOpen_iff_of_mem, ← preimage_compl, compl_compl, preimage_image_eq _ coe_injective]
| Mathlib.Topology.Compactification.OnePoint.240_0.tbRhoScp8wX1EEx | theorem isOpen_compl_image_coe {s : Set X} :
IsOpen ((↑) '' s : Set (OnePoint X))ᶜ ↔ IsClosed s ∧ IsCompact s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s✝ : Set (OnePoint X)
t s : Set X
⊢ IsClosed (some '' s) ↔ IsClosed s ∧ IsCompact s | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [← isOpen_compl_iff, isOpen_compl_image_coe] | @[simp]
theorem isClosed_image_coe {s : Set X} :
IsClosed ((↑) '' s : Set (OnePoint X)) ↔ IsClosed s ∧ IsCompact s := by
| Mathlib.Topology.Compactification.OnePoint.246_0.tbRhoScp8wX1EEx | @[simp]
theorem isClosed_image_coe {s : Set X} :
IsClosed ((↑) '' s : Set (OnePoint X)) ↔ IsClosed s ∧ IsCompact s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ IsClosed {∞} | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [← compl_range_coe, isClosed_compl_iff] | theorem isClosed_infty : IsClosed ({∞} : Set (OnePoint X)) := by
| Mathlib.Topology.Compactification.OnePoint.279_0.tbRhoScp8wX1EEx | theorem isClosed_infty : IsClosed ({∞} : Set (OnePoint X)) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ IsOpen (range some) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact isOpen_range_coe | theorem isClosed_infty : IsClosed ({∞} : Set (OnePoint X)) := by
rw [← compl_range_coe, isClosed_compl_iff]
| Mathlib.Topology.Compactification.OnePoint.279_0.tbRhoScp8wX1EEx | theorem isClosed_infty : IsClosed ({∞} : Set (OnePoint X)) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
x : X
h : NeBot (𝓝[≠] x)
⊢ NeBot (𝓝[≠] ↑x) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simpa [nhdsWithin_coe, preimage, coe_eq_coe] using h.map some | /-- If `x` is not an isolated point of `X`, then `x : OnePoint X` is not an isolated point
of `OnePoint X`. -/
instance nhdsWithin_compl_coe_neBot (x : X) [h : NeBot (𝓝[≠] x)] :
NeBot (𝓝[≠] (x : OnePoint X)) := by
| Mathlib.Topology.Compactification.OnePoint.301_0.tbRhoScp8wX1EEx | /-- If `x` is not an isolated point of `X`, then `x : OnePoint X` is not an isolated point
of `OnePoint X`. -/
instance nhdsWithin_compl_coe_neBot (x : X) [h : NeBot (𝓝[≠] x)] :
NeBot (𝓝[≠] (x : OnePoint X)) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ 𝓝[≠] ∞ = map some (coclosedCompact X) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _ | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) := by
| Mathlib.Topology.Compactification.OnePoint.308_0.tbRhoScp8wX1EEx | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) | Mathlib_Topology_Compactification_OnePoint |
case refine'_1
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ ∀ (i : Set (OnePoint X)), ∞ ∈ i ∧ IsOpen i → ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ i ∩ {∞}ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rintro s ⟨hs, hso⟩ | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) := by
refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _
· | Mathlib.Topology.Compactification.OnePoint.308_0.tbRhoScp8wX1EEx | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) | Mathlib_Topology_Compactification_OnePoint |
case refine'_1.intro
X : Type u_1
inst✝ : TopologicalSpace X
s✝ : Set (OnePoint X)
t : Set X
s : Set (OnePoint X)
hs : ∞ ∈ s
hso : IsOpen s
⊢ ∃ i', (IsClosed i' ∧ IsCompact i') ∧ some '' i'ᶜ ⊆ s ∩ {∞}ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | refine' ⟨_, (isOpen_iff_of_mem hs).mp hso, _⟩ | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) := by
refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _
· rintro s ⟨hs, hso⟩
| Mathlib.Topology.Compactification.OnePoint.308_0.tbRhoScp8wX1EEx | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) | Mathlib_Topology_Compactification_OnePoint |
case refine'_1.intro
X : Type u_1
inst✝ : TopologicalSpace X
s✝ : Set (OnePoint X)
t : Set X
s : Set (OnePoint X)
hs : ∞ ∈ s
hso : IsOpen s
⊢ some '' (some ⁻¹' s)ᶜᶜ ⊆ s ∩ {∞}ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [Subset.rfl] | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) := by
refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _
· rintro s ⟨hs, hso⟩
refine' ⟨_, (isOpen_iff_of_mem hs).mp hso, _⟩
| Mathlib.Topology.Compactification.OnePoint.308_0.tbRhoScp8wX1EEx | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) | Mathlib_Topology_Compactification_OnePoint |
case refine'_2
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ ∀ (i' : Set X), IsClosed i' ∧ IsCompact i' → ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' i'ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rintro s ⟨h₁, h₂⟩ | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) := by
refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _
· rintro s ⟨hs, hso⟩
refine' ⟨_, (isOpen_iff_of_mem hs).mp hso, _⟩
simp [Subset.rfl]
· | Mathlib.Topology.Compactification.OnePoint.308_0.tbRhoScp8wX1EEx | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) | Mathlib_Topology_Compactification_OnePoint |
case refine'_2.intro
X : Type u_1
inst✝ : TopologicalSpace X
s✝ : Set (OnePoint X)
t s : Set X
h₁ : IsClosed s
h₂ : IsCompact s
⊢ ∃ i, (∞ ∈ i ∧ IsOpen i) ∧ i ∩ {∞}ᶜ ⊆ some '' sᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | refine' ⟨_, ⟨mem_compl infty_not_mem_image_coe, isOpen_compl_image_coe.2 ⟨h₁, h₂⟩⟩, _⟩ | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) := by
refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _
· rintro s ⟨hs, hso⟩
refine' ⟨_, (isOpen_iff_of_mem hs).mp hso, _⟩
simp [Subset.rfl]
· rintro s ⟨h₁, h₂⟩
| Mathlib.Topology.Compactification.OnePoint.308_0.tbRhoScp8wX1EEx | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) | Mathlib_Topology_Compactification_OnePoint |
case refine'_2.intro
X : Type u_1
inst✝ : TopologicalSpace X
s✝ : Set (OnePoint X)
t s : Set X
h₁ : IsClosed s
h₂ : IsCompact s
⊢ (some '' s)ᶜ ∩ {∞}ᶜ ⊆ some '' sᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [compl_image_coe, ← diff_eq, subset_preimage_image] | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) := by
refine' (nhdsWithin_basis_open ∞ _).ext (hasBasis_coclosedCompact.map _) _ _
· rintro s ⟨hs, hso⟩
refine' ⟨_, (isOpen_iff_of_mem hs).mp hso, _⟩
simp [Subset.rfl]
· rintro s ⟨h₁, h₂⟩
refine' ⟨_, ⟨mem_compl in... | Mathlib.Topology.Compactification.OnePoint.308_0.tbRhoScp8wX1EEx | theorem nhdsWithin_compl_infty_eq : 𝓝[≠] (∞ : OnePoint X) = map (↑) (coclosedCompact X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : NoncompactSpace X
⊢ NeBot (𝓝[≠] ∞) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [nhdsWithin_compl_infty_eq] | /-- If `X` is a non-compact space, then `∞` is not an isolated point of `OnePoint X`. -/
instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : OnePoint X)) := by
| Mathlib.Topology.Compactification.OnePoint.318_0.tbRhoScp8wX1EEx | /-- If `X` is a non-compact space, then `∞` is not an isolated point of `OnePoint X`. -/
instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : OnePoint X)) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : NoncompactSpace X
⊢ NeBot (map some (coclosedCompact X)) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | infer_instance | /-- If `X` is a non-compact space, then `∞` is not an isolated point of `OnePoint X`. -/
instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : OnePoint X)) := by
rw [nhdsWithin_compl_infty_eq]
| Mathlib.Topology.Compactification.OnePoint.318_0.tbRhoScp8wX1EEx | /-- If `X` is a non-compact space, then `∞` is not an isolated point of `OnePoint X`. -/
instance nhdsWithin_compl_infty_neBot [NoncompactSpace X] : NeBot (𝓝[≠] (∞ : OnePoint X)) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ 𝓝 ∞ = map some (coclosedCompact X) ⊔ pure ∞ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [← nhdsWithin_compl_infty_eq, nhdsWithin_compl_singleton_sup_pure] | theorem nhds_infty_eq : 𝓝 (∞ : OnePoint X) = map (↑) (coclosedCompact X) ⊔ pure ∞ := by
| Mathlib.Topology.Compactification.OnePoint.330_0.tbRhoScp8wX1EEx | theorem nhds_infty_eq : 𝓝 (∞ : OnePoint X) = map (↑) (coclosedCompact X) ⊔ pure ∞ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ HasBasis (𝓝 ∞) (fun s => IsClosed s ∧ IsCompact s) fun s => some '' sᶜ ∪ {∞} | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [nhds_infty_eq] | theorem hasBasis_nhds_infty :
(𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
(↑) '' sᶜ ∪ {∞} := by
| Mathlib.Topology.Compactification.OnePoint.334_0.tbRhoScp8wX1EEx | theorem hasBasis_nhds_infty :
(𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
(↑) '' sᶜ ∪ {∞} | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ HasBasis (map some (coclosedCompact X) ⊔ pure ∞) (fun s => IsClosed s ∧ IsCompact s) fun s => some '' sᶜ ∪ {∞} | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact (hasBasis_coclosedCompact.map _).sup_pure _ | theorem hasBasis_nhds_infty :
(𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
(↑) '' sᶜ ∪ {∞} := by
rw [nhds_infty_eq]
| Mathlib.Topology.Compactification.OnePoint.334_0.tbRhoScp8wX1EEx | theorem hasBasis_nhds_infty :
(𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
(↑) '' sᶜ ∪ {∞} | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ comap some (𝓝 ∞) = coclosedCompact X | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [nhds_infty_eq, comap_sup, comap_map coe_injective] | @[simp]
theorem comap_coe_nhds_infty : comap ((↑) : X → OnePoint X) (𝓝 ∞) = coclosedCompact X := by
| Mathlib.Topology.Compactification.OnePoint.341_0.tbRhoScp8wX1EEx | @[simp]
theorem comap_coe_nhds_infty : comap ((↑) : X → OnePoint X) (𝓝 ∞) = coclosedCompact X | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
f : Filter (OnePoint X)
⊢ f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed s → IsCompact s → some '' sᶜ ∪ {∞} ∈ f | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp only [hasBasis_nhds_infty.ge_iff, and_imp] | theorem le_nhds_infty {f : Filter (OnePoint X)} :
f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' sᶜ ∪ {∞} ∈ f := by
| Mathlib.Topology.Compactification.OnePoint.346_0.tbRhoScp8wX1EEx | theorem le_nhds_infty {f : Filter (OnePoint X)} :
f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' sᶜ ∪ {∞} ∈ f | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
f : Ultrafilter (OnePoint X)
⊢ ↑f ≤ 𝓝 ∞ ↔ ∀ (s : Set X), IsClosed s → IsCompact s → some '' s ∉ f | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp only [le_nhds_infty, ← compl_image_coe, Ultrafilter.mem_coe,
Ultrafilter.compl_mem_iff_not_mem] | theorem ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
(f : Filter (OnePoint X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' s ∉ f := by
| Mathlib.Topology.Compactification.OnePoint.351_0.tbRhoScp8wX1EEx | theorem ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
(f : Filter (OnePoint X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' s ∉ f | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
α : Type u_2
f : OnePoint X → α
l : Filter α
⊢ Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [nhds_infty_eq, and_comm] | theorem tendsto_nhds_infty' {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ (↑)) (coclosedCompact X) l := by
| Mathlib.Topology.Compactification.OnePoint.357_0.tbRhoScp8wX1EEx | theorem tendsto_nhds_infty' {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ (↑)) (coclosedCompact X) l | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
α : Type u_2
f : OnePoint X → α
l : Filter α
⊢ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l ↔
∀ s ∈ l, f ∞ ∈ s ∧ ∃ t, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp only [tendsto_pure_left, hasBasis_coclosedCompact.tendsto_left_iff, forall_and,
and_assoc, exists_prop] | theorem tendsto_nhds_infty {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔
∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
tendsto_nhds_infty'.trans <| by
| Mathlib.Topology.Compactification.OnePoint.362_0.tbRhoScp8wX1EEx | theorem tendsto_nhds_infty {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔
∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
Y : Type u_2
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
⊢ Tendsto (f ∘ some) (coclosedCompact X) (𝓝 (f ∞)) ↔
∀ s ∈ 𝓝 (f ∞), ∃ t, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ some) tᶜ s | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp only [hasBasis_coclosedCompact.tendsto_left_iff, and_assoc] | theorem continuousAt_infty {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
ContinuousAt f ∞ ↔
∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
continuousAt_infty'.trans <| by | Mathlib.Topology.Compactification.OnePoint.375_0.tbRhoScp8wX1EEx | theorem continuousAt_infty {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
ContinuousAt f ∞ ↔
∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
Y : Type u_2
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
x : X
⊢ ContinuousAt f ↑x ↔ ContinuousAt (f ∘ some) x | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt] | theorem continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x := by
| Mathlib.Topology.Compactification.OnePoint.381_0.tbRhoScp8wX1EEx | theorem continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
Y : Type u_2
inst✝ : TopologicalSpace Y
f : OnePoint X → Y
x : X
⊢ Tendsto (f ∘ some) (𝓝 x) (𝓝 (f ↑x)) ↔ Tendsto (f ∘ some) (𝓝 x) (𝓝 ((f ∘ some) x)) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rfl | theorem continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x := by
rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]; | Mathlib.Topology.Compactification.OnePoint.381_0.tbRhoScp8wX1EEx | theorem continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : NoncompactSpace X
⊢ DenseRange some | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [DenseRange, ← compl_infty] | /-- If `X` is not a compact space, then the natural embedding `X → OnePoint X` has dense range.
-/
theorem denseRange_coe [NoncompactSpace X] : DenseRange ((↑) : X → OnePoint X) := by
| Mathlib.Topology.Compactification.OnePoint.386_0.tbRhoScp8wX1EEx | /-- If `X` is not a compact space, then the natural embedding `X → OnePoint X` has dense range.
-/
theorem denseRange_coe [NoncompactSpace X] : DenseRange ((↑) : X → OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : NoncompactSpace X
⊢ Dense {∞}ᶜ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact dense_compl_singleton _ | /-- If `X` is not a compact space, then the natural embedding `X → OnePoint X` has dense range.
-/
theorem denseRange_coe [NoncompactSpace X] : DenseRange ((↑) : X → OnePoint X) := by
rw [DenseRange, ← compl_infty]
| Mathlib.Topology.Compactification.OnePoint.386_0.tbRhoScp8wX1EEx | /-- If `X` is not a compact space, then the natural embedding `X → OnePoint X` has dense range.
-/
theorem denseRange_coe [NoncompactSpace X] : DenseRange ((↑) : X → OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
x y : OnePoint X
⊢ Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x', x = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y' | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | induction x using OnePoint.rec | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
| Mathlib.Topology.Compactification.OnePoint.419_0.tbRhoScp8wX1EEx | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' | Mathlib_Topology_Compactification_OnePoint |
case h₁
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
y : OnePoint X
⊢ Inseparable ∞ y ↔ ∞ = ∞ ∧ y = ∞ ∨ ∃ x', ∞ = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y' | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | induction y using OnePoint.rec | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
induction x using OnePoint.rec <;> | Mathlib.Topology.Compactification.OnePoint.419_0.tbRhoScp8wX1EEx | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' | Mathlib_Topology_Compactification_OnePoint |
case h₂
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
y : OnePoint X
x✝ : X
⊢ Inseparable (↑x✝) y ↔ ↑x✝ = ∞ ∧ y = ∞ ∨ ∃ x', ↑x✝ = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y' | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | induction y using OnePoint.rec | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
induction x using OnePoint.rec <;> | Mathlib.Topology.Compactification.OnePoint.419_0.tbRhoScp8wX1EEx | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' | Mathlib_Topology_Compactification_OnePoint |
case h₁.h₁
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ Inseparable ∞ ∞ ↔ ∞ = ∞ ∧ ∞ = ∞ ∨ ∃ x', ∞ = ↑x' ∧ ∃ y', ∞ = ↑y' ∧ Inseparable x' y' | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl] | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
induction x using OnePoint.rec <;> induction y using OnePoint.rec <;>
| Mathlib.Topology.Compactification.OnePoint.419_0.tbRhoScp8wX1EEx | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' | Mathlib_Topology_Compactification_OnePoint |
case h₁.h₂
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
x✝ : X
⊢ Inseparable ∞ ↑x✝ ↔ ∞ = ∞ ∧ ↑x✝ = ∞ ∨ ∃ x', ∞ = ↑x' ∧ ∃ y', ↑x✝ = ↑y' ∧ Inseparable x' y' | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl] | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
induction x using OnePoint.rec <;> induction y using OnePoint.rec <;>
| Mathlib.Topology.Compactification.OnePoint.419_0.tbRhoScp8wX1EEx | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' | Mathlib_Topology_Compactification_OnePoint |
case h₂.h₁
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
x✝ : X
⊢ Inseparable ↑x✝ ∞ ↔ ↑x✝ = ∞ ∧ ∞ = ∞ ∨ ∃ x', ↑x✝ = ↑x' ∧ ∃ y', ∞ = ↑y' ∧ Inseparable x' y' | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl] | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
induction x using OnePoint.rec <;> induction y using OnePoint.rec <;>
| Mathlib.Topology.Compactification.OnePoint.419_0.tbRhoScp8wX1EEx | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' | Mathlib_Topology_Compactification_OnePoint |
case h₂.h₂
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
x✝¹ x✝ : X
⊢ Inseparable ↑x✝¹ ↑x✝ ↔ ↑x✝¹ = ∞ ∧ ↑x✝ = ∞ ∨ ∃ x', ↑x✝¹ = ↑x' ∧ ∃ y', ↑x✝ = ↑y' ∧ Inseparable x' y' | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl] | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
induction x using OnePoint.rec <;> induction y using OnePoint.rec <;>
| Mathlib.Topology.Compactification.OnePoint.419_0.tbRhoScp8wX1EEx | theorem inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ IsCompact univ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | have : Tendsto ((↑) : X → OnePoint X) (cocompact X) (𝓝 ∞) := by
rw [nhds_infty_eq]
exact (tendsto_map.mono_left cocompact_le_coclosedCompact).mono_right le_sup_left | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ := by
| Mathlib.Topology.Compactification.OnePoint.437_0.tbRhoScp8wX1EEx | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ Tendsto some (cocompact X) (𝓝 ∞) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [nhds_infty_eq] | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ := by
have : Tendsto ((↑) : X → OnePoint X) (cocompact X) (𝓝 ∞) := by
| Mathlib.Topology.Compactification.OnePoint.437_0.tbRhoScp8wX1EEx | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
⊢ Tendsto some (cocompact X) (map some (coclosedCompact X) ⊔ pure ∞) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact (tendsto_map.mono_left cocompact_le_coclosedCompact).mono_right le_sup_left | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ := by
have : Tendsto ((↑) : X → OnePoint X) (cocompact X) (𝓝 ∞) := by
rw [nhds_infty_eq]
| Mathlib.Topology.Compactification.OnePoint.437_0.tbRhoScp8wX1EEx | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
this : Tendsto some (cocompact X) (𝓝 ∞)
⊢ IsCompact univ | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [← insert_none_range_some X] | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ := by
have : Tendsto ((↑) : X → OnePoint X) (cocompact X) (𝓝 ∞) := by
rw [nhds_infty_eq]
exact (tendsto_map.mono_left cocompact_le_coclosedCompact).mono_r... | Mathlib.Topology.Compactification.OnePoint.437_0.tbRhoScp8wX1EEx | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
this : Tendsto some (cocompact X) (𝓝 ∞)
⊢ IsCompact (insert none (range Option.some)) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact this.isCompact_insert_range_of_cocompact continuous_coe | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ := by
have : Tendsto ((↑) : X → OnePoint X) (cocompact X) (𝓝 ∞) := by
rw [nhds_infty_eq]
exact (tendsto_map.mono_left cocompact_le_coclosedCompact).mono_r... | Mathlib.Topology.Compactification.OnePoint.437_0.tbRhoScp8wX1EEx | /-- For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : T0Space X
⊢ T0Space (OnePoint X) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | refine' ⟨fun x y hxy => _⟩ | /-- The one point compactification of a `T0Space` space is a `T0Space`. -/
instance [T0Space X] : T0Space (OnePoint X) := by
| Mathlib.Topology.Compactification.OnePoint.446_0.tbRhoScp8wX1EEx | /-- The one point compactification of a `T0Space` space is a `T0Space`. -/
instance [T0Space X] : T0Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : T0Space X
x y : OnePoint X
hxy : Inseparable x y
⊢ x = y | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rcases inseparable_iff.1 hxy with (⟨rfl, rfl⟩ | ⟨x, rfl, y, rfl, h⟩) | /-- The one point compactification of a `T0Space` space is a `T0Space`. -/
instance [T0Space X] : T0Space (OnePoint X) := by
refine' ⟨fun x y hxy => _⟩
| Mathlib.Topology.Compactification.OnePoint.446_0.tbRhoScp8wX1EEx | /-- The one point compactification of a `T0Space` space is a `T0Space`. -/
instance [T0Space X] : T0Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case inl.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : T0Space X
hxy : Inseparable ∞ ∞
⊢ ∞ = ∞
case inr.intro.intro.intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : T0Space X
x y : X
h : Inseparable x y
hxy : Inseparable ↑x ↑y
⊢ ↑x = ↑y | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exacts [rfl, congr_arg some h.eq] | /-- The one point compactification of a `T0Space` space is a `T0Space`. -/
instance [T0Space X] : T0Space (OnePoint X) := by
refine' ⟨fun x y hxy => _⟩
rcases inseparable_iff.1 hxy with (⟨rfl, rfl⟩ | ⟨x, rfl, y, rfl, h⟩)
| Mathlib.Topology.Compactification.OnePoint.446_0.tbRhoScp8wX1EEx | /-- The one point compactification of a `T0Space` space is a `T0Space`. -/
instance [T0Space X] : T0Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : T1Space X
z : OnePoint X
⊢ IsClosed {z} | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | induction z using OnePoint.rec | /-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z := by
| Mathlib.Topology.Compactification.OnePoint.452_0.tbRhoScp8wX1EEx | /-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z | Mathlib_Topology_Compactification_OnePoint |
case h₁
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : T1Space X
⊢ IsClosed {∞} | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact isClosed_infty | /-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z := by
induction z using OnePoint.rec
· | Mathlib.Topology.Compactification.OnePoint.452_0.tbRhoScp8wX1EEx | /-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z | Mathlib_Topology_Compactification_OnePoint |
case h₂
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : T1Space X
x✝ : X
⊢ IsClosed {↑x✝} | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [← image_singleton, isClosed_image_coe] | /-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z := by
induction z using OnePoint.rec
· exact isClosed_infty
· | Mathlib.Topology.Compactification.OnePoint.452_0.tbRhoScp8wX1EEx | /-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z | Mathlib_Topology_Compactification_OnePoint |
case h₂
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝ : T1Space X
x✝ : X
⊢ IsClosed {x✝} ∧ IsCompact {x✝} | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact ⟨isClosed_singleton, isCompact_singleton⟩ | /-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z := by
induction z using OnePoint.rec
· exact isClosed_infty
· rw [← image_singleton, isClosed_image_coe]
| Mathlib.Topology.Compactification.OnePoint.452_0.tbRhoScp8wX1EEx | /-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
⊢ T4Space (OnePoint X) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_sup_right, nhds_coe_eq, coclosedCompact_eq_cocompact,
disjoint_map coe_injective, ← principal_singleton, disjoint_principal_right, compl_infty]
exact ⟨disjoint_nhds_cocompact z, range_mem_map⟩ | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
| Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
z : X
⊢ Disjoint (𝓝 ↑z) (𝓝 ∞) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rw [nhds_infty_eq, disjoint_sup_right, nhds_coe_eq, coclosedCompact_eq_cocompact,
disjoint_map coe_injective, ← principal_singleton, disjoint_principal_right, compl_infty] | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
| Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
z : X
⊢ Disjoint (𝓝 z) (cocompact X) ∧ range some ∈ map some (𝓝 z) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact ⟨disjoint_nhds_cocompact z, range_mem_map⟩ | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
⊢ T4Space (OnePoint X) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | suffices : T2Space (OnePoint X) | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
this : T2Space (OnePoint X)
⊢ T4Space (OnePoint X)
case this
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLoca... | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | infer_instance | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case this
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
⊢ T2Space (OnePoint X) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | refine t2Space_iff_disjoint_nhds.2 fun x y hxy => ?_ | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case this
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
x y : OnePoint X
hxy : x ≠ y
⊢ Disjoint (𝓝 x) (𝓝 y) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | induction x using OnePoint.rec | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case this.h₁
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
y : OnePoint X
hxy : ∞ ≠ y
⊢ Disjoint (𝓝 ∞) (𝓝 y) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | induction y using OnePoint.rec | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case this.h₂
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
y : OnePoint X
x✝ : X
hxy : ↑x✝ ≠ y
⊢ Disjoint (𝓝 ↑x✝) (𝓝 y) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | induction y using OnePoint.rec | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case this.h₁.h₁
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
hxy : ∞ ≠ ∞
⊢ Disjoint (𝓝 ∞) (𝓝 ∞) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact (hxy rfl).elim | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case this.h₁.h₂
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
x✝ : X
hxy : ∞ ≠ ↑x✝
⊢ Disjoint (𝓝 ∞) (𝓝 ↑x✝) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact (key _).symm | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case this.h₂.h₁
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
x✝ : X
hxy : ↑x✝ ≠ ∞
⊢ Disjoint (𝓝 ↑x✝) (𝓝 ∞) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | exact key _ | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
case this.h₂.h₂
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : WeaklyLocallyCompactSpace X
inst✝ : T2Space X
key : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)
x✝¹ x✝ : X
hxy : ↑x✝¹ ≠ ↑x✝
⊢ Disjoint (𝓝 ↑x✝¹) (𝓝 ↑x✝) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | rwa [nhds_coe_eq, nhds_coe_eq, disjoint_map coe_injective, disjoint_nhds_nhds,
← coe_injective.ne_iff] | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := by
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z => by
rw [nhds_infty_eq, disjoint_s... | Mathlib.Topology.Compactification.OnePoint.460_0.tbRhoScp8wX1EEx | /-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
instance [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
⊢ ¬Continuous ⇑CofiniteTopology.of.symm | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | inhabit X | /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous. -/
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm := by
| Mathlib.Topology.Compactification.OnePoint.481_0.tbRhoScp8wX1EEx | /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous. -/
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X
⊢ ¬Continuous ⇑CofiniteTopology.of.symm | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simp only [continuous_iff_continuousAt, ContinuousAt, not_forall] | /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous. -/
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm := by
inhabit... | Mathlib.Topology.Compactification.OnePoint.481_0.tbRhoScp8wX1EEx | /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous. -/
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm | Mathlib_Topology_Compactification_OnePoint |
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X
⊢ ∃ x, ¬Tendsto (⇑CofiniteTopology.of.symm) (𝓝 x) (𝓝 (CofiniteTopology.of.symm x)) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | use CofiniteTopology.of ↑(default : X) | /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous. -/
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm := by
inhabit... | Mathlib.Topology.Compactification.OnePoint.481_0.tbRhoScp8wX1EEx | /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous. -/
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm | Mathlib_Topology_Compactification_OnePoint |
case h
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X
⊢ ¬Tendsto (⇑CofiniteTopology.of.symm) (𝓝 (CofiniteTopology.of ↑default))
(𝓝 (CofiniteTopology.of.symm (CofiniteTopology.of ↑default))) | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using
(finite_singleton ((default : X) : OnePoint X)).infinite_compl | /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous. -/
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm := by
inhabit... | Mathlib.Topology.Compactification.OnePoint.481_0.tbRhoScp8wX1EEx | /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous. -/
theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm | Mathlib_Topology_Compactification_OnePoint |
V : Type u_1
inst✝¹ : Category.{?u.28, u_1} V
inst✝ : Abelian V
X Y Z : V
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ g.op ≫ f.op = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← op_comp, w, op_zero] | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by | Mathlib.Algebra.Homology.Opposite.40_0.Q6AwB66K8LWeNih | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{?u.28, u_1} V
inst✝ : Abelian V
X Y Z : V
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ f ≫ factorThruImage g = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w, zero_comp] | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by | Mathlib.Algebra.Homology.Opposite.40_0.Q6AwB66K8LWeNih | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{u_2, u_1} V
inst✝ : Abelian V
X Y Z : V
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ imageToKernel g.op f.op (_ : g.op ≫ f.op = 0) =
(imageSubobjectIso g.op ≪≫ (imageOpOp g).symm).hom ≫
(cokernel.desc f (factorThruImage g) (_ : f ≫ factorThruImage g = 0)).op ≫
(kernelSubobjectIs... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | ext | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... | Mathlib.Algebra.Homology.Opposite.40_0.Q6AwB66K8LWeNih | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... | Mathlib_Algebra_Homology_Opposite |
case h
V : Type u_1
inst✝¹ : Category.{u_2, u_1} V
inst✝ : Abelian V
X Y Z : V
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ imageToKernel g.op f.op (_ : g.op ≫ f.op = 0) ≫ Subobject.arrow (kernelSubobject f.op) =
((imageSubobjectIso g.op ≪≫ (imageOpOp g).symm).hom ≫
(cokernel.desc f (factorThruImage g) (_ : f ≫ fact... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | simp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelOpOp_inv, Category.assoc,
imageToKernel_arrow, kernelSubobject_arrow', kernel.lift_ι, ← op_comp, cokernel.π_desc,
← imageSubobject_arrow, ← imageUnopOp_inv_comp_op_factorThruImage g.op] | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... | Mathlib.Algebra.Homology.Opposite.40_0.Q6AwB66K8LWeNih | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... | Mathlib_Algebra_Homology_Opposite |
case h
V : Type u_1
inst✝¹ : Category.{u_2, u_1} V
inst✝ : Abelian V
X Y Z : V
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ (imageSubobjectIso g.op).hom ≫ (imageUnopOp g.op).inv ≫ (factorThruImage g.op.unop).op =
(imageSubobjectIso g.op).hom ≫ (imageOpOp g).inv ≫ (factorThruImage g).op | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rfl | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... | Mathlib.Algebra.Homology.Opposite.40_0.Q6AwB66K8LWeNih | theorem imageToKernel_op {X Y Z : V} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.op f.op (by rw [← op_comp, w, op_zero]) =
(imageSubobjectIso _ ≪≫ (imageOpOp _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w,... | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{?u.11101, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ g.unop ≫ f.unop = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← unop_comp, w, unop_zero] | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by | Mathlib.Algebra.Homology.Opposite.53_0.Q6AwB66K8LWeNih | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{?u.11101, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ f ≫ factorThruImage g = 0 | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w, zero_comp] | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by | Mathlib.Algebra.Homology.Opposite.53_0.Q6AwB66K8LWeNih | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... | Mathlib_Algebra_Homology_Opposite |
V : Type u_1
inst✝¹ : Category.{u_2, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ imageToKernel g.unop f.unop (_ : g.unop ≫ f.unop = 0) =
(imageSubobjectIso g.unop ≪≫ (imageUnopUnop g).symm).hom ≫
(cokernel.desc f (factorThruImage g) (_ : f ≫ factorThruImage g = 0)).unop ≫
... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | ext | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... | Mathlib.Algebra.Homology.Opposite.53_0.Q6AwB66K8LWeNih | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... | Mathlib_Algebra_Homology_Opposite |
case h
V : Type u_1
inst✝¹ : Category.{u_2, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ imageToKernel g.unop f.unop (_ : g.unop ≫ f.unop = 0) ≫ Subobject.arrow (kernelSubobject f.unop) =
((imageSubobjectIso g.unop ≪≫ (imageUnopUnop g).symm).hom ≫
(cokernel.desc f (factorThruImag... | /-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additiv... | dsimp only [imageUnopUnop] | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... | Mathlib.Algebra.Homology.Opposite.53_0.Q6AwB66K8LWeNih | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... | Mathlib_Algebra_Homology_Opposite |
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